Test 3 Review
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     Part I ( Multiple choice)

     1. Use the ratio test to determine whether the series 
        is absolutely convergent, conditionally convergent, or divergent.

        WW. 8.4, 1- 5
 
         |a_{n+1}|/|a_n| --> s < 1  ==> the series is convergent.
              n --> infinity

     2. Use the geometric series 

         (1/(1-x))^(-1)= 1 + x + x^2 + x^3 + . . .+ x^n  + ...

        for |x| < 1  
      to write a power series representation for a function  

 
        WW. 8.6, 1-3

     3. Find the Taylor series of f(x) about x = a. 

        WW. 8.7,  6



     4. Find the largest interval of convergence of the series 

       WW. 8.5, 4, 6, 9  


     5. Use known MacLaurin series to find the MacLaurin series of f(x).


       WW. 8.6, 5, 6 

     6. Use MacLaurin series for f(x) (family of 1/(1-x)) to obtain the MacLaurin series for the function f(x). It is a theoretical question about how to get the power representation for ln(1+x) (and arctan(x)) from 1/(1+x).
                 
        WW. 8.6, 5, 6, 8   
   (it is about ln(1+x) and arctan(x) but not in the same form as on the test) 


      7. Eliminate the parameter to find the Cartesian equation for a curve.
   
        WW. 9.1, 1 


     8. Parametrization of a circle (ellipse)  

        WW.  9.1, 2, 3, 5, 9, 10

     9. For a parametric curve defined by x(t), y(t) find
        the slope of the line tangent to the curve at the point
        corresponding to t=a

        WW. 9.2, 2


     10. For a parametric curve defined by x(t), y(t) find
         all values of t where the curve has a vertical(horizontal)
         tangent line.

        WW. 9.2, 3

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         Part II (Free response)
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   1. Let f(x)=...  Find the Taylor series about a. (You will need to find the first 4 non-zero terms)
            
         WW. 8.7, 7       


   2. What is the largest set of convergence? (Use interval notations)
       
       WW.8.5, 4, 6     

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Bonus problem (Taylor series or power series)